New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. (English) Zbl 0947.42024

Given a window function \(\psi \in L^2(\mathbb R)\), the Gabor transform of a function \(f\in L^2(\mathbb R)\) is given by \(Gf(\omega,t)= \frac{1}{\sqrt{2\pi}} \int_\infty^\infty f(x) \overline{\psi(x-t)}e^{-i\omega x} dx\). It is proved that if \(f\neq 0\), then the support of \(Gf\) has infinite Lebesgue measure. A similar result is stated for the wavelet transform. An abstract framework including both cases is presented.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
43A32 Other transforms and operators of Fourier type
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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